Online Talk: George Drummond

YouTube recording:

Time: Thursday, July 6, 3pm ET

Speaker: George Drummond, University of Canterbury
Title: Elastic elements and where to find them

Abstract: In this talk we will discuss the presence of “elastic” elements in 3-connected matroids. An element e of a 3-connected matroid M is elastic if both the simplification of M/e and the cosimplification of M\e are 3-connected. Those familiar with Bixby’s Lemma will recall that at least one of these two conditions holds for every element. We will consider the obstructions to the presence of elastic elements, namely 4-element fans and another family of 3-separating structures we dub “theta-separators”, before arriving at our wheels-and-whirls-type theorem for elastic elements. In particular, it is shown that if neither of the aforementioned obstructions is present, then we are guaranteed at least four elastic elements. This wheels-and-whirls-type result then extends naturally to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified 3-connected minor. We will also consider some applications of these results to the related study of removing elements relative to a fixed basis.

Online Talk: Tung Nguyen

YouTube recording:

Time: Thursday, June 22, 3pm ET

Speaker: Tung Nguyen, Princeton University
Title: More graphs with the Erdős–Hajnal property

Abstract: Erdős and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $n$-vertex graph with no induced copy of $H$ contains a clique or stable set of size at least $n^c$. Alon, Pach, and Solymosi reduced this conjecture to the case when $H$ is prime, or equivalently when $H$ is not a blow-up of smaller graphs. Until now, it was not known for any prime $H$ on at least six vertices. This talk describes a construction of infinitely many prime graphs $H$ each satisfying the Erdős–Hajnal conjecture. The proof method actually gives the stronger result that every such $H$ is viral, which will be explained in the talk. Joint work with Alex Scott and Paul Seymour.